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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field and let $X \\subseteq \\mathbb{P}^n_k$ be a [[projective variety|A4H4CZ2V35]] of [[dimension|36X2WTHYCB]] $d$.\nLet $I(X) \\subseteq S = k[x_0, \\dots, x_n]$ be its homogeneous ideal, and let $P_X$ be the [[Hilbert polynomial|RMH4FBCFDA]] of the graded ring $S/I(X)$.\nThen $P_X$ has degree $d$ and leading coefficient $e/d!$ for a positive integer $e$.\nThe *degree* of $X$ is $\\deg X = e$.\n\n## Remarks\n\nEquivalently, $\\deg X$ is the number of points of $X \\cap L$ for a general linear subspace $L \\subseteq \\mathbb{P}^n$ of dimension $n - d$.\n\nExample: let $X = V(f) \\subseteq \\mathbb{P}^2$ with $f$ an irreducible homogeneous polynomial of degree $e$.\nThe exact sequence $0 \\to S(-e) \\to S \\to S/(f) \\to 0$ gives $P_X(t) = \\binom{t+2}{2} - \\binom{t-e+2}{2} = et - e(e-3)/2$.\nHere $d = 1$ and the leading coefficient is $e$, so $\\deg X = e$.\n\nNon-example of the geometric form: the conic $X = V(xz - y^2)$ has degree $2$.\nThe line $y = 0$ meets it in the two points $[1:0:0]$ and $[0:0:1]$.\nThe tangent line $x = 0$ meets it only in $[0:0:1]$.\nThe tangent line is not general, so it does not compute the degree; it counts the point with multiplicity $2$.\n\nThe degree depends on the embedding: $\\mathbb{P}^1$ embedded as a line has degree $1$, and embedded as the conic above it has degree $2$.\nFor a hypersurface $V(f)$ with $f$ irreducible, the degree of the variety is the degree of $f$.\n",
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